A particle moves with the velocity `vec v = (5 vec i + 2 vec j - vec k) ms^(-1)` under the influence of a constant force, `vec F = (2 vec i + 5 vec j - 10 vec k)N`. The instantaneous power applied is

Options

A. 5 W
B. `10 W
C. 20 W
D. 30 W

Explanation

To find the instantaneous power applied to the particle, we can use the formula for power in terms of force and velocity:

### Step 1: Understand the formula for instantaneous power
The instantaneous power \( P \) can be calculated using the formula:
\[ P = \vec{F} \cdot \vec{v} \]
where \( \vec{F} \) is the force vector and \( \vec{v} \) is the velocity vector.

### Step 2: Write down the given vectors
From the problem, we have:
\[ \vec{v} = (5 \hat{i} + 2 \hat{j} - 1 \hat{k}) \, \text{m/s} \]
\[ \vec{F} = (2 \hat{i} + 5 \hat{j} - 10 \hat{k}) \, \text{N} \]

### Step 3: Calculate the dot product
Now, we will calculate the dot product \( \vec{F} \cdot \vec{v} \):
\[ \vec{F} \cdot \vec{v} = (2 \hat{i} + 5 \hat{j} - 10 \hat{k}) \cdot (5 \hat{i} + 2 \hat{j} - 1 \hat{k}) \]
Using the properties of dot product:
\[ = (2 \cdot 5) + (5 \cdot 2) + (-10 \cdot -1) \]
Calculating each term:
\[ = 10 + 10 + 10 \]
\[ = 30 \]

### Step 4: State the final answer
Thus, the instantaneous power applied is:
\[ P = 30 \, \text{W} \]